Average Error: 7.9 → 0.9
Time: 14.2s
Precision: 64
\[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -9.805921840000082361388853030105062603537 \cdot 10^{80} \lor \neg \left(y \le 1.909280761605855696977057249980191966985 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]
\frac{\cosh x \cdot \frac{y}{x}}{z}
↓
\begin{array}{l}
\mathbf{if}\;y \le -9.805921840000082361388853030105062603537 \cdot 10^{80} \lor \neg \left(y \le 1.909280761605855696977057249980191966985 \cdot 10^{-68}\right):\\
\;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{\frac{x}{y}}}{z}\\

\end{array}
double f(double x, double y, double z) {
        double r340458 = x;
        double r340459 = cosh(r340458);
        double r340460 = y;
        double r340461 = r340460 / r340458;
        double r340462 = r340459 * r340461;
        double r340463 = z;
        double r340464 = r340462 / r340463;
        return r340464;
}
↓
double f(double x, double y, double z) {
        double r340465 = y;
        double r340466 = -9.805921840000082e+80;
        bool r340467 = r340465 <= r340466;
        double r340468 = 1.9092807616058557e-68;
        bool r340469 = r340465 <= r340468;
        double r340470 = !r340469;
        bool r340471 = r340467 || r340470;
        double r340472 = x;
        double r340473 = cosh(r340472);
        double r340474 = r340473 * r340465;
        double r340475 = z;
        double r340476 = r340475 * r340472;
        double r340477 = r340474 / r340476;
        double r340478 = 0.5;
        double r340479 = -r340472;
        double r340480 = exp(r340479);
        double r340481 = exp(r340472);
        double r340482 = r340480 + r340481;
        double r340483 = r340478 * r340482;
        double r340484 = r340472 / r340465;
        double r340485 = r340483 / r340484;
        double r340486 = r340485 / r340475;
        double r340487 = r340471 ? r340477 : r340486;
        return r340487;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.9
Target0.4
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -4.618902267687041990497740832940559043667 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{elif}\;y \lt 1.038530535935153018369520384190862667426 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x} \cdot \cosh x\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -9.805921840000082e+80 or 1.9092807616058557e-68 < y

    1. Initial program 20.6

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]
    2. Using strategy rm

    3. Applied associate-*r/20.6

      \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z}\]

    4. Applied associate-/l/0.8

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}}\]

    if -9.805921840000082e+80 < y < 1.9092807616058557e-68

    1. Initial program 0.8

      \[\frac{\cosh x \cdot \frac{y}{x}}{z}\]

    2. Taylor expanded around inf 10.7

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{1}{2} \cdot e^{x} + \frac{1}{2} \cdot e^{-x}\right)}{x \cdot z}}\]

    3. Simplified0.9

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{\frac{x}{y}}}{z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -9.805921840000082361388853030105062603537 \cdot 10^{80} \lor \neg \left(y \le 1.909280761605855696977057249980191966985 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{\cosh x \cdot y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(e^{-x} + e^{x}\right)}{\frac{x}{y}}}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.03853053593515302e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))